The quadratic isoperimetric inequality for mapping tori of free group automorphisms II: The general case

If F is a finitely generated free group and \phi is an automorphism of F then the mapping torus of \phi admits a quadratic isoperimetric inequality. This is the third and final paper in a series proving this theorem. The first two were math.GR/0211459 and math.GR/0507589.Comment: 73 page

Existence and conditional energetic stability of solitary gravity-capillary water waves with constant vorticity

We present an existence and stability theory for gravity-capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimiser of the wave energy $\mathcal H$ subject to the constraint $\mathcal I=2\mu$, where $\mathcal I$ is the wave momentum and $0< \mu \ll 1$. Since $\mathcal H$ and $\mathcal I$ are both conserved quantities a standard argument asserts the stability of the set $D_\mu$ of minimisers: solutions starting near $D_\mu$ remain close to $D_\mu$ in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are described by solutions of a Korteweg-deVries equation (for strong surface tension) or a nonlinear Schr\"{o}dinger equation (for weak surface tension). We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the appropriate model equation as $\mu \downarrow 0$Comment: Corrected version. To appear in Proceedings of the Royal Society of Edinburgh: Section

A dimension-breaking phenomenon for water waves with weak surface tension

It is well known that the water-wave problem with weak surface tension has small-amplitude line solitary-wave solutions which to leading order are described by the nonlinear Schr\"odinger equation. The present paper contains an existence theory for three-dimensional periodically modulated solitary-wave solutions which have a solitary-wave profile in the direction of propagation and are periodic in the transverse direction; they emanate from the line solitary waves in a dimension-breaking bifurcation. In addition, it is shown that the line solitary waves are linearly unstable to long-wavelength transverse perturbations. The key to these results is a formulation of the water wave problem as an evolutionary system in which the transverse horizontal variable plays the role of time, a careful study of the purely imaginary spectrum of the operator obtained by linearising the evolutionary system at a line solitary wave, and an application of an infinite-dimensional version of the classical Lyapunov centre theorem.Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s00205-015-0941-

Fully Localised Three-Dimensional Gravity-Capillary Solitary Waves on Water of Infinite Depth

Fully localised three-dimensional solitary waves are steady water waves which are evanescent in every horizontal direction. Existence theories for fully localised three-dimensional solitary waves on water of finite depth have recently been published, and in this paper we establish their existence on deep water. The governing equations are reduced to a perturbation of the two-dimensional nonlinear Schr¨odinger equation, which admits a family of localised solutions. Two of these solutions are symmetric in both horizontal directions and an application of a suitable variant of the implicit-function theorem shows that they persist under perturbations

A bifurcation theory for three-dimensional oblique travelling gravity-capillary water waves

This article presents a rigorous existence theory for small-amplitude three-dimensional travelling water waves. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable. Wave motions which are periodic in a second, different horizontal direction are detected using a centre-manifold reduction technique by which the problem is reduced to a locally equivalent Hamiltonian system with a finite number of degrees of freedom. A catalogue of bifurcation scenarios is compiled by means of a geometric argument based upon the classical dispersion relation for travelling water waves. Taking all parameters into account, one finds that this catalogue includes virtually any bifurcation or resonance known in Hamiltonian systems theory. Nonlinear bifurcation theory is carried out for a representative selection of bifurcation scenarios; solutions of the reduced Hamiltonian system are found by applying results from the well-developed theory of finite-dimensional Hamiltonian systems such as the Lyapunov centre theorem and the Birkhoff normal form. We find oblique line waves which depend only upon one spatial direction which is not aligned with the direction of wave propagation; the waves have periodic, solitary-wave or generalised solitary-wave profiles in this distinguished direction. Truly three-dimensional waves are also found which have periodic, solitary-wave or generalised solitary-wave profiles in one direction and are periodic in another. In particular, we recover doubly periodic waves with arbitrary fundamental domains and oblique versions of the results on threedimensional travelling waves already in the literature

The mass-metallicity relation of local active galaxies

We systematically measure the gas-phase metallicities and the mass-metallicity relation of a large sample of local active galaxies for the first time. Observed emission-line fluxes from the Sloan Digital Sky Survey (SDSS) are compared to a four-dimensional grid of photoionization models using the Bayesian parameter estimation code NebulaBayes. For the first time we take into account arbitrary mixing between HII region and narrow-line region (NLR) emission, and the models are also varied with metallicity, ionization parameter in the NLR, and the gas pressure. The active galactic nucleus (AGN) oxygen abundance is found to increase by $\Delta {\rm O/H} \sim 0.1$ dex as a function of host galaxy stellar mass over the range $10.1 < \log M_* / M_\odot < 11.3$. We also measure the metallicity and ionization parameter of 231000 star-forming galaxies for comparison with the sample of 7670 Seyfert 2 galaxies. A systematic offset in oxygen abundance of 0.09 dex is observed between the mass-metallicity relations of the star-forming and active galaxies. We investigate potential causes of the offset, including sample selection and the treatment in the models of diffuse ionized gas, pressure, and ionization parameter. We cannot identify the major cause(s), but suspect contributions due to deficiencies in modeling the ionizing spectra and the treatment of dust physics. Optical diagnostic diagrams are presented with the star-forming and Seyfert data colored by the inferred oxygen abundance, ionization parameter and gas pressure, clearly illustrating the trends in these quantities.Comment: 12 pages, 4 figures and 1 table; accepted for publication in Ap

Interrogating Seyferts with NebulaBayes: Spatially probing the narrow-line region radiation fields and chemical abundances

NebulaBayes is a new Bayesian code that implements a general method of comparing observed emission-line fluxes to photoionization model grids. The code enables us to extract robust, spatially resolved measurements of abundances in the extended narrow line regions (ENLRs) produced by Active Galactic Nuclei (AGN). We observe near-constant ionization parameters but steeply radially-declining pressures, which together imply that radiation pressure regulates the ENLR density structure on large scales. Our sample includes four `pure Seyfert' galaxies from the S7 survey that have extensive ENLRs. NGC2992 shows steep metallicity gradients from the nucleus into the ionization cones. An {\it inverse} metallicity gradient is observed in ESO138-G01, which we attribute to a recent gas inflow or minor merger. A uniformly high metallicity and hard ionizing continuum are inferred across the ENLR of Mrk573. Our analysis of IC5063 is likely affected by contamination from shock excitation, which appears to soften the inferred ionizing spectrum. The peak of the ionizing continuum E_peak is determined by the nuclear spectrum and the absorbing column between the nucleus and the ionized nebula. We cannot separate variation in this intrinsic E_peak from the effects of shock or HII region contamination, but E_peak measurements nevertheless give insights into ENLR excitation. We demonstrate the general applicability of NebulaBayes by analyzing a nuclear spectrum from the non-active galaxy NGC4691 using a HII region grid. The NLR and HII region model grids are provided with NebulaBayes for use by the astronomical community.Comment: Accepted for publication in ApJ; 29 pages with 10 figures and 3 table

Low Apparent Survival and Heterogeneous Movement Patterns of Invasive Blue Catfish in a Coastal River

Blue Catfish Ictalurus furcatus were purposefully introduced into freshwater tributaries to Chesapeake Bay in the past, and populations have subsequently spread to new areas, negatively impacting native communities and causing concern for resource managers. To aid development of management strategies, we implemented a multiyear (2012-2015) tagging study of invasive Blue Catfish in a 40-km stretch of the Potomac River to estimate survival and assess movement patterns. Blue Catfish (N = 1,237) were captured by electrofishing and double-tagged to allow us to estimate tag retention rates; we used reward tags to increase reporting rates. Recaptured fish (N = 104; 8.4% return rate) were at large for between 2 and 1,208 d. Tag retention rates were 0.88 (SE = 0.045) after 1 year and declined to 0.31 (SE = 0.107) after 2.7 years. The mean minimum distance moved by fish was 24.1 km (range = 0.0-112.6 km). Most (63%) fish displayed downriver movements, but distance moved was unrelated to fish size or days at large. Greater distances were observed among fish that moved downriver (34.4 km) than those that moved upriver (6.7 km). These results suggest high variability in movement behaviors for Blue Catfish inhabiting the tidal Potomac River from freshwater reaches to estuarine habitats. We estimated an annual apparent survival rate of 0.56 (SE = 0.057; Brownie tag-return model) across the study period. This survival rate is lower than survival rates reported from their native range. Long-distance movements of Blue Catfish in the Potomac River indicate that robust, large-scale control measures will be needed to reduce population abundance and minimize negative impacts of this species on native communities

Transverse instability of gravity-capillary line solitary water waves

The gravity-capillary water-wave problem concerns the irrotational flow of a perfect fluid in a domain bounded below by a rigid bottom and above by a free surface under the influence of gravity and surface tension. In the case of large surface tension the system has a travelling line solitary-wave solution for which the free surface has a localised profile in the direction of propagation and is homogeneous in the transverse direction. In this note we show that this line solitary wave is linearly unstable under spatially inhomogeneous perturbations which are periodic in the direction transverse to propagation

Stress-Energy Tensor for the Massless Spin 1/2 Field in Static Black Hole Spacetimes

The stress-energy tensor for the massless spin 1/2 field is numerically computed outside and on the event horizons of both charged and uncharged static non-rotating black holes, corresponding to the Schwarzschild, Reissner-Nordstrom and extreme Reissner-Nordstr\"om solutions of Einstein's equations. The field is assumed to be in a thermal state at the black hole temperature. Comparison is made between the numerical results and previous analytic approximations for the stress-energy tensor in these spacetimes. For the Schwarzschild (charge zero) solution, it is shown that the stress-energy differs even in sign from the analytic approximation. For the Reissner-Nordstrom and extreme Reissner-Nordstrom solutions, divergences predicted by the analytic approximations are shown not to exist.Comment: 5 pages, 4 figures, additional discussio